# Geometric Explanation of Tamagawa Numbers

Sometimes in order to understand a concept thoroughly we need to have a algebraic view ( in terms of equations ) and corresponding geometric view.

My interest always lies with understanding the Tamagawa Numbers from different view points. So to say something, I already know the algebraic version of Tamagawa numbers ( both local and global parts ) . It can be stated as follows " the semisimple linear algebraic group $G$ over $\mathbb Q$, the Tamagawa number of $G$ (which is the standard terminology for the volume of $G(\mathbb Q)\setminus G(\mathbb A)$ with respect to Tamagawa measure) should be equal to $1$, and when $G$ is an elliptic curve rather than a linear group, there are many interesting things that happened, and which gave rise to a series of seminal works, and the Tamagawa numbers are widely used for tori ( by T.Ono ) , and also later the Birch and Swinnerton-Dyer found an analogue of the tamagawa number of elliptic curve ( anlogous to the work of T.Ono in defining the tamagawa numbers of Tori ) that played a central part in defining the Birch and Swinnerton-Dyer conjectures."

In terms of measure the Tamagawa Number can be defined as $$\large \tau=\rho(G)^{-1} |\Delta_k|^{-\large \frac{1}{2} \rm { dim } G } \prod_{\nu \ | \infty} \omega_{\nu} \prod_{\mathfrak{p}} L_{\mathfrak{p}}(1,\chi_{G})\omega_{\mathfrak{p}}$$ where $\omega$ is the gauge form on $G$, $\chi_{G}$ is the character of the representation of the Galois group of $\bar{k}/k$ on the lattice $\widehat{G}$ and $\rho(G) = \lim_{ s \mapsto 1 } (s - 1)^{r} L(s, \chi_{G})$ , $\Delta_k$ be the discriminant of $k/\mathbb{Q}$. And in some sense $$\large \prod_{\nu \ | \infty} \omega_{\nu} \prod_{\mathfrak{p}} L_{\mathfrak{p}}(1,\chi_{G})\omega_{\mathfrak{p}}$$ can be taken as the Haar Measure on $G(\mathbb{A})$.

I am now very curious in hearing to alternate interpretations of Tamagawa Number in Geometric sense, as a intuitive way. If there is some geometric way of explaining what is the tamagawa number , and what impact does it give , I would be very happy in listening that. Please don't include the wikipedia article ( as I have read it already ).

P.S : I know the answer will express something related to the differential forms, but I am a bit confused. If there is some other intuitive explanation , I would be very happy. Even any article that does this job will be fine.

Thank you.

• I already know the differential geometric version of it. – IDOK Apr 30 '12 at 16:16
• By the way, have you posted the question on MathOverFlow? Per chance you could get better answers? – awllower Feb 5 '13 at 16:57
• – Watson Mar 9 '18 at 18:59

## 1 Answer

Here are my two cents: some references 1 and 2.

• The first paper is the one, which I have already got after searching in google, and the second one I don't know frankly. But I assume that if I had not seen the google and searched about this, your articles might have helped me a lot. So , why to wait, I will award a bounty to it ( even though the answer completely satisfy my requirements, at-least I must thank you for your helping nature ). I will wait for some time, and see if some one can add something, or else I award the bounty to you, instead of wasting it. – IDOK May 3 '12 at 5:29
• Those were some easy $\geq 535$ points. =) – Rasmus May 4 '12 at 14:36
• @Rasmus : Yes, but what to do, I didn't have any other alternative. No body cared to answer the question, so that was only the way, to save the bounty from getting wasted. At-least some body enjoy the reputation. – IDOK May 5 '12 at 10:13
• @Iyengar, sorry for the lateness. My answer was just to save the bounty from waste. I hope you found it useful. Thanks for the bounty. – Mathfun May 15 '12 at 8:15
• None of the two links works... The first one is THE GEOMETRY OF TAMAGAWA NUMBERS OF CHEVALLEY GROUPS by KAI BEHREND AND AJNEET DHILLON. The second one is On Tamagawa Numbers by Takashi Ono. – Watson Mar 9 '18 at 18:51